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The task was to find out the value of $$\sin\frac{31\pi}3$$

This is a example in my book in which following steps are shown: $$\begin{align} \sin\frac{31\pi}3& =\sin\left(10\pi +\frac\pi3\right)\\ &=\sin\frac\pi3\\ &=\frac{\sqrt3}2 \end{align}$$

I cannot understand step 3, $=\sin\frac\pi3$

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    $\begingroup$ $$ \sin(n\pi+\theta)=\left\{ \begin{array}{l l} \sin\theta & ;\quad \text{if $n$ is even},\\ \\ -\sin\theta & ;\quad \text{if $n$ is odd}. \end{array} \right. $$ $\endgroup$
    – Tunk-Fey
    May 28, 2014 at 14:34
  • $\begingroup$ I think all answers are good. But i think we can accept only one.Is there any way by which i can accept more than one ? $\endgroup$
    – Freddy
    May 28, 2014 at 14:49
  • $\begingroup$ @Tunk-Fey you are right $\endgroup$
    – Freddy
    May 28, 2014 at 14:58
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    $\begingroup$ Old math joke: The value of sin? Value==wages and biblically, the wages of sin is death, so the value of sin==death. Try that one on your math teacher. $\endgroup$
    – TechZen
    May 28, 2014 at 17:09
  • $\begingroup$ @TechZen i am definetly going to try this on my maths teacher $\endgroup$
    – Freddy
    May 28, 2014 at 17:45

5 Answers 5

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The sine function is periodic with period $2\pi$. This means that $\sin\theta = \sin(\theta + 2\pi) \ \forall \ \theta$. One can apply this identity five times to get rid of the $10 \pi$ term.

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    $\begingroup$ What? 8 upvotes? How did that happen!? $\endgroup$
    – Aidan F. Pierce
    May 28, 2014 at 21:48
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    $\begingroup$ By the time you have read my last comment, the number 8 will have become obsolete. $\endgroup$
    – Aidan F. Pierce
    May 28, 2014 at 22:04
  • $\begingroup$ Really...really...11 upvotes on this question $\endgroup$
    – user3001408
    May 29, 2014 at 9:35
  • $\begingroup$ @user3001408 This is answer :) $\endgroup$
    – Freddy
    Oct 10, 2014 at 5:56
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The sine function is periodic with period $2\pi$. A periodic function is a function that repeats its values in regular intervals or periods. A function is said to be periodic with period $P$, where $P$ being a nonzero constant if we have $$ f(x+P) = f(x) $$ for all values of $x$ in the domain. Therefore $$ \sin(n\pi+\theta)=\left\{ \begin{array}{l l} \sin\theta & ;\quad \text{if $n$ is even},\\ \\ -\sin\theta & ;\quad \text{if $n$ is odd}. \end{array} \right. $$

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  • $\begingroup$ Unfortunately the upvotes do not increase my reputation right now. :( $$\\$$Thanks for the upvoters. :) $\endgroup$
    – Tunk-Fey
    May 28, 2014 at 14:41
  • $\begingroup$ I think later it will be added up :) $\endgroup$
    – Freddy
    May 28, 2014 at 14:54
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Multiples of $2\pi$ do not matter because $\sin$ is periodic of period $2\pi$.

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    $\begingroup$ The key fact is that sine is $2\pi$-periodic, and not simply periodic. $\endgroup$
    – Fly by Night
    May 28, 2014 at 14:42
  • $\begingroup$ All answer are good but i am choosing this as it is shortest one $\endgroup$
    – Freddy
    May 28, 2014 at 14:59
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Think of the unit circle and the definition of $\sin$: A straight line drawn from the origin with the angle between the $x$-axis of $\theta$ intersects the unit circle at $(\cos \theta, \sin \theta)$.

unit circle

Say you have a point on the unit circle at $(\cos \theta, \sin \theta)$. If the line making the angle $\theta$ from the origin makes a full revolution (revolves around the origin by a measure of $2\pi$), then it will end up in the same position. Thus, $\sin n = \sin (2\pi + n) = \sin (n - 2\pi)$.

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You can confirm this by using the angle-sum formula:

$$\begin{align} \sin(10\pi + \pi/3) & = \sin(10\pi)\cos(\pi/3) + \cos(10 \pi)\sin(\pi/3) \\ \\ & = 0\cdot\frac 12 + 1 \cdot \sin (\pi/3) \\ \\ & = \sin(\pi/3)\\ \\ & = \sqrt 3/2\end{align}$$

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